Defining polynomial
|
\(x^{4} + 2 x + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $4$ |
|
| Ramification index $e$: | $4$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $4$ |
|
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_1$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\frac{4}{3}, \frac{4}{3}]$ | |
| Visible Swan slopes: | $[\frac{1}{3},\frac{1}{3}]$ | |
| Means: | $\langle\frac{1}{6}, \frac{1}{4}\rangle$ | |
| Rams: | $(\frac{1}{3}, \frac{1}{3})$ | |
| Jump set: | $[1, 2, 5]$ | |
| Roots of unity: | $2$ |
|
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
|
| Relative Eisenstein polynomial: |
\( x^{4} + 2 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $24$ |
| Galois group: | $S_4$ (as 4T5) |
| Inertia group: | $A_4$ (as 4T4) |
| Wild inertia group: | $C_2^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3}]$ |
| Galois mean slope: | $1.1666666666666667$ |
| Galois splitting model: | $x^{4} - 2 x + 2$ |